The Binomial and Poisson Distributions

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Edexcel IAL Statistics 2 (WST02)

The Binomial and Poisson Distributions

The big idea: A binomial counts successes in a fixed number of independent trials; a Poisson counts random events in a fixed interval — and for large n with small p, one approximates the other.

Summary — What This Chapter Covers

X ~ B(n, p): P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ, mean np, variance np(1 − p).
X ~ Po(λ): P(X = r) = e^−λλʳ/r!, mean = variance = λ.
Poisson conditions: events occur singly, independently, at a constant average rate.
B(n, p) ≈ Po(np) when n is large and p is small.

1. The Binomial Distribution

Binomial

Fixed trials, constant p

Check the conditions in context: fixed number of trials, two outcomes, independent trials, constant probability.

Worked Example

X ~ B(10, 0.3). Find P(X = 2).

2. The Poisson Distribution

Poisson

Events in an interval

Scale λ to the interval: 3 per hour is Po(1.5) for 30 minutes.

Worked Example

X ~ Po(2.5). Find P(X ≤ 2).

3. Approximating a Binomial by a Poisson

Approximation

Large n, small p

Use Po(np) when n > 50 and np < 5 (a guideline).

Practice Question

X ~ B(200, 0.01). Use a Poisson approximation to find P(X ≤ 1).

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The Binomial and Poisson Distributions